Part 2
The first Line therefore HI is call’d the Ground-line, or Line of the Plan, at which the Edifice begins, and on which it stands. The second Line NON, parallel to the former, is call’d the Horizontal Line, wherein is plac’d O the Point of the Eye, and N the Point of the Distance. Two Points of Distance are here laid down, that you may make use of which you please; for that on one Side only is sufficient for the fore-short’ning Figures in Perspective: Neither can any Optick Delineation, or Perspective, be described, without first making two Parallels; one of the Plan, or Ground-line, the other of the Horizon; marking, in the Line of the Horizon, the Point of the Eye, or Sight, and the Point of Distance. It was thought besides expedient to put one and the same Thing into three Schemes or Designs, to let you see, that the Place, from which the Figure C is to be look’d upon, is the Point N, one of the right Lines NO, which must be conceived as fixt at right Angles into O; the Distance ON being the same as that between A and DE in the Plan, or between B and GF in the Upright.
In Pictures taking up a great deal of Room, the Point of Sight ought to be made in the middle of the Horizontal Line; and where the Height of the Picture happens to be greater than the Breadth, the Distance NO must be made equal to the Height. If the Breadth of the Picture exceed the Height, the Distance NO must be made equal to the Breadth: For so will the Extent of the Picture be the better comprehended, or receiv’d, at one View. And altho’ the same Distance may seem to be used in a different manner in the Plan A, and in the Elevation B, from what it is in C; nevertheless the Sections of the visual Rays, with the Wall of the Plan A, and of the Elevation B, have a perfect Correspondence with the Sections of those of the Figure C.
Now, if to the Spectator in A and B, we would have the farthest Part of the Work seem to recede from the Lines DE and GF, as much as the Square P does, whose Elevation is Q; draw from the Points A and B, the visual Rays to the extreme Points of the Square P and Q; noting the Sections they make with the Walls DE and GF; which by some is call’d the Veil, Transparent Medium, Section, Cloth, or Table: and you’ll find RS equal to TV, XZ equal to YK; and so of the rest.
FIGURA Secunda.
Modus delineandi opticè Quadratum.
_Ante descriptionem opticam quadrati ~A~, quod fingimus delineatum esse in papyro separatâ, ducendæ sunt duæ lineæ parallelæ; altera plani, altera horizontis, ut jam docuimus; notando in linea horizontis punctum oculi ~O~, & punctum distantiæ ~E~. Tum translatâ in lineam plani latitudine ac longitudine ipsius quadrati ~A~, ita ut linea ~CB~ sit æqualis latitudini, & ~DC~ sit æqualis longitudini. Ex punctis ~B~ & ~C~ fiunt visuales ~BO~, ~CO~ ad punctum oculi; ex puncto ~D~ fit recta ~DE~ ad punctum distantiæ. Demum ubi visualem ~CO~ secat recta ~DE~, fit ~GF~ parallela ad ~CB~; habesque quadratum opticè contractum._
_Compendium temporis & laboris facies, præsertim in schematibus quæ abundant lineis, si chartulam in medio complicaveris, eademque utaris, ut latitudinem ac longitudinem quadrati transferas in lineam plani._
The Second Figure.
_Manner of delineating a ~Square~ in Perspective._
Before the _Square_ A, which is supposed to be drawn on a separate Paper, can be laid down in Perspective, two parallel Lines must be drawn; one of the Plan, the other of the Horizon, as is already intimated; noting in the Horizontal Line the Point of Sight O, and the Point of Distance E. Then, when the Length and Breadth of the Square A shall be transferr’d into the Line of the Plan, so that the Line CB be equal to the Breadth, and DC be equal to the Length, let the visual Lines BO, CO be drawn from the Points B and C to the Point of Sight O, and the right Line DE from the Point D to the Point of Distance. Lastly, where the Line DE cuts the Visual CO, make GF parallel to CB: and you have the _Square_ Optically contracted, or fore-shorten’d in Perspective.
To spare Time and Pains, especially in Figures that abound in Lines, fold your Paper in the middle, and make use of it to transfer the Breadth and Length of the Square, into the Line of the Plan.
FIGURA TERTIA.
Optica delineatio rectanguli, alterâ parte longioris.
_Latitudo ~BC~ rectanguli ~A~ ponatur in linea plani, adhibito circino, vel chartulâ complicatâ; & ex punctis ~B~ & ~C~ fiant visuales ad ~O~, punctum perspectivæ. Tum papyro ex altera parte iterum complicatâ, notetur longitudo ~CD~ rectanguli; ducendo tum rectam ~DE~ ad punctum distantiæ, tum rectam ~FG~ parallelam ad ~BC~, quæ complebit opticam delineationem rectanguli._
_Altera figura ostendit complicationem cruciformem papyri, quæ adhiberi potest in delineandis rectangulis, seu latitudo eorum sit major longitudine, aut vice versâ; seu latitudo & longitudo sint æquales._
The Third Figure.
_The Delineation of an Oblong ~Square~ in ~Perspective~._
Let the Breadth BC of the Square A, be plac’d in the Line of the Plan, by the Compass, or a folded Paper, and from the Points B and C, make the Visuals to the Point of Sight O. Then fold your Paper cross-wise, and mark CD the Length of the Square, drawing the Line DE to the Point of Distance, and the Line FG parallel to BC, which will complete the Optick Delineation of the oblong Square.
The other Figure shews the Folding of the Paper cross-wise, which is of ready use in delineating Squares, whose Breadth exceeds their Length, or _vice versâ_; or whose Length and Breadth are equal.
FIGURA QUARTA.
Optica descriptio quadrati duplicis.
_Iam incipies frui compendio papyri complicatæ. Nam eam admovendo lineæ plani, nullo negotio notare poteris puncta ~1~, ~2~, ~3~, ~4~, ~5~, ~6~, linearum visualium, quæ ducentur ad ~O~ punctum perspectivæ. Exinde complicatâ rursum chartulâ in crucem ad ~P~, notabuntur hæc puncta; ~7~, coincidens cum puncto ~6~, nisi quadratum distet à linea plani; ~8~, ~9~, ~10~. Ductis autem rectis ex ~8~, ~9~, ~10~, ad punctum ~E~, ubi secant visualem ~6~, ~7~, ~O~ fient parallelæ, eritque completa delineatio._
_In medio quadrati ~B~, aliud quadratum facilè describetur, ducendo diagonales seu diametros ab angulo ad angulum, ut in figura._
The Fourth Figure.
_The Optical Delineation of a double Square._
Here you’ll find the Advantage of your folded Paper; for, applying it to the Line of the Plan, you readily mark the Points 1, 2, 3, 4, 5, 6, of the visual Lines, which must be drawn to the Point of Sight O. Then folding the Paper cross-wise, as in P, you mark the Points 7, 8, 9, 10, placing the Point 7 on that of 6, unless you would have the Square removed within the Line of the Plan. Then from 8, 9, 10, drawing Lines to the Point of Distance E; where they intersect the Line 6, 7, O, draw Parallels to the Line of the Plan; and your Work is done.
Within the Square B, you may easily inscribe another Square, by help of the Diagonals; as may be seen in the Figure.
FIGURA QUINTA.
Vestigia quadratorum, cum elevationibus.
_Suppositis iis quæ jam diximus de Contractione optica Quadratorum, notandum est, vestigium primi Quadrati distare à linea plani spatio ~BA~ opticè contracto; quia linea ~BD~ habet à visuali ~AO~, distantiam ~BA~. Eodem modo Quadratum secundum distat à linea plani spatio ~EA~, & sic deinceps._
_Velim observes, in omnibus his Quadratis lineas longitudinis esse partes visualium, lineas vero latitudinis esse parallelas lineæ plani, & in primo Quadrato duci ex punctis, in quibus lineæ ~BD~, ~CD~, tendentes ad punctum distantiæ, secant visualem ~AO~._
_Sub singulis vestigiis Quadratorum, delineavimus alia omnino similia, per quæ parvo labore fient tres bases, erigendo ad libitum duas primas perpendiculares æquales; ac ducendo tum duas visuales ad punctum oculi ~O~, tum reliquas, ut in figura. Supponendum est autem, geometricam altitudinem cujuslibet rei desumi ex lineis normalibus ad lineam plani; quemadmodum latitudo & longitudo geometrica desumuntur ex eadem linea plani._
_Tres aliæ bases inferiores formantur sine lineis occultis ex vestigio & ex elevatione longitudinis opticè deformatis, adhibendo solas altitudines ac latitudines angulorum. Nomine altitudinis intelligimus distantiam cujuslibet anguli à linea plani; nomine latitudinis intelligimus distantiam anguli ab una aliqua linea normali ad lineam plani; dummodo hæ normales eandem habeant positionem respectu basium, & respectu vestigiorum & elevationum. Quemadmodum autem per concursum altitudinis ~FG~, & latitudinis ~HI~, ope duorum circinorum invenitur unus angulus in una basi; ita inveniuntur cæteri tum in ea, tum in reliquis._
The Fifth Figure.
_Plans of Squares, with their Elevations._
Besides what has been already said of the fore-short’ning of Squares in Perspective, it is convenient to observe, That the Foot of the first Square is here set within the Line of the Plan, as much as the Space BA optically contracted; because the Line BD has the Distance BA from the Visual AO: And in like manner, the second Square is distant from the Line of the Plan the Space EA; and so for the rest.
I would have you observe in all these Squares, That by the _Length_ I always understand part of the visual Lines, and by the _Breadth_ those parallel to the Ground-line; which in the first Square are drawn from the Points in which the Lines BD, CD, tending to the Point of Distance, intersect the Visual AO.
Under the Plans of these Squares are described three others just like them, which are easily converted into three Bases, by erecting, at pleasure, the two first Perpendiculars of equal Height, and thence drawing two Visuals to the Point of Sight O, which also bound the rest, as in the Figure. Observe also, That the Geometrical Height of every thing is to be set perpendicularly from the Ground-line, or Line of the Plan, as the Geometrical Length and Breadth are also placed on the same Line.
The three other Bases below are form’d without the Help of Occult Lines, by making use only of the Heights and Breadths of the Angles, taken from the Perspective Plan and Upright. By _Height_ I understand the Distance of each Angle, or Corner, from the Ground-Line; By _Breadth_, the Distance of an Angle, or Corner, from any Line perpendicular to the Ground-line; provided these Lines have always the same Place in respect of the Bases, as they have in respect of the Perspective Plan and Upright. And as, by the Help of two Compasses, the Height FG, and the Breadth HI determine the Corner of the first Base; so, in like manner, are found the Corners of the other Bases.
FIGURA SEXTA.
Modus opticæ delineationis, absque lineis occultis.
_In hac figura sexta, vestigium geometricum ~B~ seorsim posui ab elevatione geometrica ~A~, ut deinceps faciemus. Vestigium ~B~ opticè contractum in ~E~ est ~NMRS~; elevatio contracta longitudinis vestigii est ~FTSN~. Posito autem quòd altitudines ~FN~, ~1, 5~, ~2, 6~, sint æquales; latitudines ~NM~, ~1, 2~, ~5, 6~, sint æquales; & rectæ ~NM~, ~5,6~, sint in linea ~X~ plani; rectæ ~FN~, ~1, 5~, sint in perpendiculo ~V~: anguli ~3~ & ~4~ basis ~C~ habent eandem elevationem seu distantiam à linea ~X~ plani, quam habet angulus ~T~: anguli ~1~ & ~2~ habent elevationem, quam angulus ~F~: anguli ~3~ & ~7~ habent eandem latitudinem seu distantiam à perpendiculo ~V~, quam habet angulus ~R~: anguli ~2~ & ~6~ habent eandem latitudinem, quam habet angulus ~M~._
The Sixth Figure.
_The Manner of designing in Perspective, without occult Lines._
In this sixth Figure, I have design’d the Geometrical Plan B separately from the Geometrical Elevation A, as I shall always do hereafter. The Plan B optically contracted, or put in Perspective, in E, is NMRS; the Elevation of its Length in Perspective is FTSN. Then supposing the Heights FN, 1,5, 2,6, equal; and the Breadths NM, 1,2, 5,6, equal; the Lines NM, 5,6, to be in the Line of the Plan X; and the Lines FN, 1,5, in the Perpendicular V: the Angles 3 and 4 of the Base C have the very same Elevation or Distance from the Line of the Plan X, as has the Angle T: the Angles 1 and 2 have the same Elevation with the Angle F: the Angles 3 and 7 have the same Breadth or Distance from the Perpendicular V, as the Angle R has: the Angles 2 and 6 have the same Breadth, as the Angle M has.
FIGURA SEPTIMA.
Aliud exemplum vestigii geometrici, cum elevatione longitudinis.
_Si delineanda sit basis dissecta in quatuor partes, fiat vestigium ~A~ cum suis divisionibus longitudinis ~ED~ & latitudinis ~CD~. Easdem vero divisiones latitudinis habebit in ~EF~ elevatio ~B~ quæ pertingit usque ad ~X~. Porro ad contractionem opticam vestigii adhibebitur papyrus complicata in latum & in longum, transferendo in lineam plani latitudinem & longitudinem vestigii. Deinde nullo negotio fiet optica deformatio elevationis, ut clarè positum est in figura. Quomodo autem ex vestigio & ex elevatione longitudinis opticè imminutis eruatur basis nitida sine lineis occultis, ex præcedentibus manifestum est. Optarem ut per assiduam circini tractationem in hac methodo exercenda operam sedulò ponas; quum ex ea pendeat omnis facilitas delineationum opticarum._
The Seventh Figure.
_Another Example of a Geometrical Plan and Upright, put in Perspective._
For drawing in Perspective a Pedestal, or Base, divided into four Parts, make the Plan A with its Divisions of Length ED, and of Breadth CD; and the same Divisions of Breadth EF, in the Elevation B, prolong’d to X. Then make the Perspective-Plan, by transferring the Breadth and Length into the Ground-line, by means of your Paper folded cross-wise. From which Plan the Perspective-Upright is very easily made, as may be plainly seen in the Figure. How the Base below, without occult Lines, is made from the Perspective-Plan and Upright, is manifest from what has been said before. I could wish you would be very diligent in the Practice of this Method by the Compass; because the Dispatch of Perspective-Delineations chiefly depends thereon.
FIGURA OCTAVA.
Optica projectio stylobatæ.
_Si libitum fuerit delineare stylobatam, cum projecturis in summo & imo, incipies ab elevatione geometrica ~A~, ducendo occultas ad id necessarias, tum versus perpendicularem ~L~, tum deorsum pro vestigio geometrico ~B~, cujus distantiæ transferentur in spatium ~G~. Si mensuræ longitudinis distent spatio ~C~ à mensuris latitudinis, vestigium deformatum videbitur distare à linea ~K~ plani, quantum est idem spatium ~C~. In construenda optica elevatione ~D~, visuales ex punctis lineæ ~L~ dabunt lineas latitudinis; lineas vero altitudinis accipies ex lineis vestigii contracti, ut in figura. In formando stylobata nitido ~EF~, locum anguli ~H~ dabit concursus latitudinis ex linea ~L~ usque ad ~M~, & altitudinis ex linea ~K~ usque ad ~I~. Concursus tum ejusdem altitudinis, tum latitudinis ex ~L~ usque ad ~O~, dabit angulum ~N~. Demum altitudinem anguli ~P~ accipies ex ~K~ usque ad ~Q~; latitudinem ex ~L~ usque ad ~R~._
The Eighth Figure.
_The Projection of a Pedestal in Perspective._
If you would draw a Pedestal, with the Projecture of its Cap and Base, you must begin with the Geometrical Elevation A, by drawing such occult Lines as are necessary, as well sideways to the Perpendicular L, as downwards for making the Geometrical Plan B, whose Distances must be transferr’d, and carry’d into the Space G. If the Measures of the Length be placed the Distance of the Space C, from those of the Breadth, the Perspective-Plan will then appear removed within the Ground-line K, as much as the said Space C is. In the Construction of the Perspective Elevation D, the Visuals drawn from the Points of the Line L give the Lines of the Breadth; and those of the Height are taken from the Lines of the Perspective-Plan, as in the Figure. In delineating the clean or finish’d Pedestal EF, the Intersection of the Breadth from L to M, with the Height from K to I, gives the precise Place of the Corner H. The Intersection of the same Height with the Breadth LO gives the Angle N. Lastly, the Angle P is found by the Intersection of the Height KQ, with that of the Breadth LR.
FIGURA NONA.
Optica delineatio Architecturæ Jacobi Barozzii; & primum, de Stylobata Ordinis Etrusci.
_Perspectiva nusquam clariùs emicat, quàm in Architectura. Iccirco tibi ob oculos pono Architecturam Jacobi Barozzii, quem à patria nuncupant ~Il Vignola~, reliquis fortasse usitatiorem; in eaque continetur elevatio geometrica singulorum quinque Ordinum, qui vocantur, Etruscus, Doricus, Ionicus, Corinthius, & Romanus, vel Compositus; delineando seorsim partes cujuscunque Ordinis in figuris grandioribus. Elevationi geometricæ suum vestigium nos addemus; ex vestigio autem & ex elevatione opticè deformatis, eliciemus apparentias solidorum juxta regulam traditam. Exempli gratia, si delineare velis stylobatam quadratum & pilam Ordinis Etrusci, præter elevationem geometricam ~A~ delineare oportet vestigium geometricum ~B~; ex ambobus autem opticè contractis formatur stylobata nitidus ~D~, cum anta & pila existente ad latus, accipiendo altitudines à linea plani, latitudines à linea perpendiculari ad ipsum planum. In alia delineatione posuimus pilam ex adverso, ut eis omni modo delineandis assuescas._
_Ad vitandam confusionem linearum, proderit ut figuræ fiant his nostris multò grandiores: in quem finem singulis paginis apposita est scala modulorum. Hoc nomine intelliguntur partes æquales, in quas dividuntur lineæ latitudinis & altitudinis elevationum geometricarum; ac lineæ latitudinis & longitudinis vestigiorum geometricorum. Si moduli sint parvi, subdividuntur singuli in duodecim partes; ac prout fuerint grandiores, subdividuntur in partes triginta, vel sexaginta, vel centumviginti. ~Modulos Etruscum Doricúmque in partes duodecim; reliquos autem in octodecim partiti sunt.~_
The Ninth Figure.
_The Architecture of ~Vignola~ in Perspective; and first, of his Pedestal of the ~Tuscan~ Order._
Perspective never appears more graceful, than in Architecture; for which Reason I present you with that of _James Barozzi_, from his Country generally call’d _Vignola_; which perhaps is more in use than any other; and contains the Geometrical Upright of each of the five Orders, _viz._ the _Tuscan_, _Dorick_, _Ionick_, _Corinthian_, and the _Roman_, or _Composite_; together with a separate Delineation of the Parts of each Order, in larger Figures. To this Geometrical Elevation we shall add the Plan, and, from both of them reduc’d into Perspective, shall draw the Appearances of Solids, according to the Rule before laid down. For Example: If you would draw the square _Tuscan_ Pedestal, and its Pilaster, you must, from the Geometrical Elevation A, make the Geometrical Plan B; and from both of them reduc’d in Perspective, draw the finish’d Pedestal D, with that of its Pilaster on the Side, by taking the Heights from the Ground-line, and the Breadths from a Line perpendicular to the same. On the other Side we have placed the Pilaster on the Back-part, that you may practise the Drawing them in any manner.
For avoiding the Confusion of Lines, I advise you to make the Figures as much larger than ours as you can; for which purpose there is annex’d a Scale of Modules to each Figure. By this Name we understand the equal Parts, into which the Lines of the Breadth and Height of the Geometrical Uprights, and of the Breadth and Length of the Geometrical Plans, are divided. If the Modules are small, they are subdivided into twelve Parts; and according as they are larger, into thirty, sixty, or an hundred and twenty Parts. _I have divided the ~Tuscan~ and ~Dorick~ Module into twelve Parts, and that of the other Orders into eighteen._
FIGURA DECIMA.
Optica deformatio stylobatæ Dorici; ubi de modo vitandi confusionem, in vestigiis delineandis.
_Elevatio geometrica ~B~ stylobatæ Dorici continet eandem symmetriam partium quæ habetur apud Barozzium; ex eaque eruitur vestigium geometricum ~A~ per lineas occultas, quæ descendant ex punctis terminativis præcipuarum projecturarum. Earundem projecturarum distantiæ transferendæ sunt in lineam elevationis, notando puncta quæ necessaria sunt ad deformandam elevationem longitudinis stylobatæ._
_Si ob propinquitatem lineæ plani ad lineam horizontis, vestigium evadat confusum, fiant in distantia congrua sub linea plani aliæ lineæ planorum ipsi parallelæ, cum suis vestigiis. Quid autem emolumenti afferat distantia major præ minori, ostendit vestigium ~E~ distinctiùs vestigio ~D~. Singula hæc vestigia fiunt notando in linea cujuslibet plani mensuras latitudinis & longitudinis vestigii ~A~, & ducendo lineas ad eadem puncta oculi ac distantiæ._
_Stylobatam nitidum descripsimus ex parte ~G~, tum ex necessitate, tum ut videas, pro distantia ~FO~, usurpandam esse distantiam ~GO~ penitus æqualem._
The Tenth Figure.
_A ~Dorick~ Pedestal in Perspective; with the Manner of avoiding Confusion, in designing the Plans._
The Geometrical Elevation B has the same Members and Proportions, as the _Dorick_ Pedestal of _Vignola_; and the Geometrical Plan A is form’d, by letting fall occult Lines from the principal Projectures of the Upright. Occult Lines are also to be continued to the Perpendicular F, from the several Members requisite for elevating in Perspective the Length of the Pedestal.
When, by reason of the too near Approach of the Ground-line to that of the Horizon, the Plan becomes thereby confus’d; draw at a convenient Distance underneath, other Ground-lines parallel to the first; together with the Plans in Perspective. And of what Advantage the Removal of the Ground-line is, is evident from the Plan E, which is much more distinct than the Plan D. Each of these Plans is made, by marking upon its respective Ground-line the Measures of the Breadth and Length of the Plan A, and by drawing Lines to the same Points of Sight and Distance, which were first assign’d.
We have placed the finish’d Pedestal on the Side G, partly for want of Room, and partly to shew, that the Point of Distance G is there made use of, GO being equal to FO.
FIGURA Undecima.
Stylobatæ Ionici deformatio; ubi de vitanda confusione in elevationibus.
_Tum in figura præcedenti, tum rursus in hac, ostendimus quid agendum sit ubi vestigia ~AA~ nimium obliquentur, unde oritur confusio; præcipuè in lineis parallelis quæ exhibent latitudines. Non minor difficultas interdum occurret in elevationibus longitudinis opticè deformandis; quòd videlicet, ob nimiam earum obliquitatem, pervium non sit altitudines singularum projecturarum probè discernere ac designare. Ad scopulos istos declinandos, loco elevationis ~B~ adhibebitur elevatio ~C~, quæ distinctior est, tum illâ, tum duabus intermediis ~D~ & ~E~, ob majorem distantiam quam habet à puncto oculi._
_In delineando stylobata nitido, latitudines accipientur ex ultimo vestigio, ponendo unam cuspidem circini in linea perpendiculari, quæ proxima est literæ ~O~: altitudines accipientur ex elevatione ~C~, ponendo unam cuspidem circini in linea plani, ut in præcedentibus ostensum est._
The Eleventh Figure.
_The ~Ionick~ Pedestal in Perspective; with the Manner of avoiding Confusion, in Elevations._